The DF Structure Models for Options Pricing On the Dividend-Paying and Capital-Splitting

Transcription

1 h F ucu Mols fo Opions Picing On h iin-paying an Capial-pliing Fng AI pamn of Managmn cinc Zhngzhou Infomaion Engining Unisiy P.O.Bo Zhngzhou Hnan 45 China fngai@public.zz.ha.cn; fngai@6.com Absac. Bas on h F sucu mols fo opion picing F. ai 5 his pap iscusss fuh h F sucu mols on h cass i.. h unlying sock bing iin-pai capial-spli o iin-pai an capial-spli. hs h cass a iscuss spaaly an a inga o h gnal mols fo call o pu. Finally h ampls a gin o compa h opions pics calcula by h F fomulas an Black-chols fomulas an hy show as a whol ha h F fomulas a no infio o Black-chols fomulas. F fomula is usful o as in financial mak bcaus i is connin o ajus along wih h aing im. Ky Wos. F sucu mol opions picing iin-paying capial-spliing Inoucion Af h opions a kins of impoan iai poucs is occu h sachs is picing mol a pai anion y much an spcially h Amican pu opion. In h suis of opion picing h ha bn many significan suls F. Black M. chols 973; R.C. Mon 976; W.F. hap 978 an appoimaion mhos fo Amican pu opion H.E. Johnson 983; R. Gsk an H.E. Johnson 984; L.W. MacMillan 986; R..aplon an M.G. ubahmanyam 997. No ac analyic fomula has bn pouc fo h alu of an Amican pu opion unil 5. h sucu mol fo opions picing is gin by F. ai A]. I coul pic any of opions inclu Amican pu opion. h nw poblms a h way how h opions b pic af iin-paying an capial spliing on is unlings incluing Euopan an Amican call o pu opions bcaus h sucu mol fo opions picing F. ai 5 is on a non-iin-paying sock. Bas on h sucu mol fo opions picing his pap will gi h ways o pic h opions af iin-paying an capial spliing on is unlings. Finally som of h mpiical ampls a off. By hs ampls w shall s h mols gin in his pap a suiabl o pic h opions on unlings wih iin-paying an capial spliing an y connin in applicaion. h basic assumpions abou sock pic h basic assumpions w us o fin an unlying sock an sock inics pic ga as h basis of h iscussion in his pap a as follows: h a pics h cos pic an h mak pic o an unlying ass. h cos pic mans h aag alu of all h pics pai by h mak as o buy an unlying ass an h mak pic is h cun chang pic of an unlying ass. h pics cos pic an mak pic ha bn flucuaing wih im. Any pic is non-ngai an h flucuaion ang i.. h aianc of pic is posii. Boh h cos pic an h flucuaion of cos pic of an unlying a h basic lmns which min h mak pics of h unlying sock. h mak pics com ino bing on h mak chang. h possibiliis ha h mak pic of unlying is much low han h cos pic o is much high han h cos pic will b y small. h a no ansacion coss o as in aing. h cos pic n no o b coninuous an h mak pics a isibu coninuously W shall pf fing o sock insa of ass a h sam im h sock can also b plac by sock inics an w mnion no mo in h following iscussions.

2 3 Paial isibuion an F sucu Fis w gi som basic finiions as follow: finiion h Paial isibuion. L b a non-ngai sochasic aiabl an i follows h isibuion of nsiy µ µ σ σ f < hn is sai o saisfy h paial isibuion an nos Pµ σ. h paial isibuion is a kin of unca nomal isibuion. finiion h Paial Pocss If sochasic aiabl is la o im i.. w ha Pµσ hn h { } is call a paial pocss. In gnal h sock pic changs along wih h im hfo w ha Assumpion L µ b h cos pic of sock a h im an σ b h aianc of cos pic a h im. If h mak pics of sock follow h basic assumpions in scion hus suppos ha h mak pic aiabl follows h paial isibuion a im an nos Pµ σ. can b ga as a sock o h mak pic of h sock. Fom 6] w ha h following hom an hom : hom L h mak pic aiabl of a sock follow h paial isibuion Pµ σ hus h pc alu E of h aag of mak pic is as follows µ µ µ σ σ σ E µ σ µ σ wh R σ is h aag aing pofi. h aianc of h mak pic aiabl h isk of h mak pic is as follows σ E µ E ] 3 hom Fo any ] h following quaions a coc appoimaly:. uµ σ u σ µ µ σ σ sgn µ. > wh sgn ; < Essnially h paial isibuion scibs sock pics in is isibuion consucion. finiion 3 F pocss If {ξ } is a sochasic pocss an ξ Pµ σ hn {ξ } is call a F pocss. Fom fnc 7] w ha finiion 4 L a an b b non-ngai consans If a>b w fin: a b a z z lim. finiion 5 F sucu L b h alu of an ass la o sock Pµ σ if an > P ]- i.. follows h nsiy isibu as f <

3 hn w call h F sochasic sucu of on. is call a F sucu of fo sho. Whn. o h F sucu mans a sochasic alu which will b qual o h cash ass in h fuu im un no ins a. Alhough h sock has cain conncions wih F sucu in aianc hi sochasic momns may ha no iniabl laion so w ha Assumpion L b h F sucu of on suppos ha an a inpnn on wih anoh. 4 h pics isibuion of unlying asss af iinpaying an capial-spliing τ. W ha a sock Pµ σ h sock is pai iins o pan capial p sha a h im 4. h pics isibuion of sock on iin-paying an non-capial-spliing If paying iins a im τ hn sock pic is Pµ σ wh µ µ σ σ τ. Af paying iins h sock pic is low bu h flucuaion spa i.. h aianc of sock pic is inaiabl. 4. h pics isibuion of sock on non-iin-paying an capial-spliing If non-iin-paying an capial-spliing a τ hn sock pic is P µ σ wh µ kµ σ k σ an k m m is h shas bfo capial-spliing an m is h m shas af capial-spliing τ. Af spliing capial h sock pic is low accoing o a of capial-spliing an h aianc of sock pic cus own accoing o h a of capial-spliing. 4.3 h pics isibuion of sock on iin-paying an capial-spliing If paying iins an capial-spliing a τ hn sock pic is P µ σ wh µ k µ σ k σ m is h Quaniy of iins paying p sha k m is h shas bfo capial-spliing m is h m shas af capial-spliing τ. Af paying iins an spliing capials h sock pic is obiously low an fuh h aianc of sock pic cus own accoing o a of capial-spliing. 5 h F ucu Mols Fo Opions Picing On A Non-iin- Paying an Non-Capial-pliing ock Accoing o fnc 7] w ha h following noaions: h mak pic of sock a h cun im h sik pic of opion. F sochasic sucu of on. cun im. 3

4 4 im of piaion of opion. isk-f a of ins o mauiy. - fowa alu of ˆ E h pc alu in a isk-nual. F sochasic sucu of on -. C alu of call opion o buy on sha. P alu of pu opion o sll on sha. If Pµσ an P - ]- w ha h F sucu mols of opions picing F mol fo sho as follows: 5. h mol fo call opion picing h pic of call opion a im is ] ma E C ] f ] ] ] sgn ] ] ] ] 4 Whn h call opion is bough fowa o cu a any im τ ] h pic of unlying sock τ bcoms a consan o h opion conac hus τ]. Acco- ing o 4 an finiion 4 h cun alu of h opion is C ττ- --τ if τ> --τ ; C τ if τ --τ ; i.. C τma{τ- --τ }. A his im h ininsic alu of h call opion is ma {τ- } hus C τ ma{τ- } 5 5. h mol fo pu opion picing h pic of pu opion a im is ] ma E P ] f ] ] ] sgn ] ] ] 6 Whn h pu opion is bough fowa o cu a any im τ ] h pic of unlying sock τ bcoms a consan o h opion conac hus τ]. Accoing o pssion 6 an finiion 4 h cun alu of h opion is P τ --τ -τ if τ< --τ P τ if τ --τ i.. P τ ma{ --τ -τ }. A his im h ininsic alu of h call opion is ma{-τ }

5 5 hus P τ ma{-τ } 7 h pssion 4 can fo sho b win as: C ] 8 An h pssion 6 can b win as: P ] 9 wh ] ]. 6 h ucu Mols Fo Opions Picing On A iin-paying an Capial-pliing ock H w suppos ha h pic of oiginal sock is Pµ σ. Af paying iins o spliing capials h pic of sock is P σ µ wh µ an σ coul b ons in scion o 4.3. F sochasic sucu of on - is ] P. noing ] an Combining fomula 4 an fomula 6 wih hom w know af paying iins o spliing capials a τ h pics of opions on sock a 6. h picing mol fo call opion on h iin-paying an capial-spliing sock h pic of call opion a im >τ ] ma E C ] f sgn 6. h picing mol fo pu opion on h iin-paying an capial-spliing sock h pic of pu opion a im >τ ] ma E P ] f

6 sgn 6.3 h gnal mols fo opions picing ] umming up h iscussions abo an supposing h iin-paying an capial-spliing sock is a im τ <τ< w ha h gnal mols fo opions picing as follow. h mol fo call opion picing is C < < τ C 3 C τ < wh is h cun im C is min by fomula 4 o 8 C is min by fomula. h mol fo pu opion picing is P < < τ P 4 P τ < wh is h cun im P is min by fomula 6 o 9 P is min by fomula. 7 h mpiical Eampls H w gi an ampl on h sock of MF MICROOF CP an is opions in fnc 7]. 7. h fiing of paial isibuion fo MF W ak h clos pics of MF as sampl aa. im: Jan. 9 -c. 4. h sima suls of paams a as follows: µˆ ; ˆ σ ; h cosponing hisogam sampls folgam an fiing cu of Paial isibuion a shown in h figu. 7. h compaai analysis fo MF im: c. 5. Pouc: h opion conac on MF. Mauiy: Epis Af: Fi6-Jan-4. Unlying: Clos pic of MF a cun a 53.39$. In abl h a h pics on c.4 which w h closing pics a acually in h Uni as opion mak P h call an pu opions pics calcula by F sucu fomulas F an h call an pu opions pics calcula by Black-chols fomulas B-. Accoing o h aa fom abl i is ifficul o know whh h F fomula is b han B- fomula o no w shoul o fuh mpiical sach. aking h sik pic 6 an fo abl. Conas of opions pics of MF ik Call opions pics Pu opions pics pics P F B- P F B

7 ampl h aiy of call an pu opion pics calcula by F fomulas an B- fomulas a spcily shown in figu a an b. 7.3 h compaisons bwn opions pics bfo an af iin-paying an capial-spliing H a h h ampls abou compaisons bwn opions pics bfo an af hi unlying sock a iin-pai an capial-spli. h cun pic of sock is h clos pic on c. 5 i $. Combining h h cass in scion spaaly w ha Compaisons bwn opions pics bfo an af h unlying sock bing iin-pai. uppos sik pic of opion is 6 iin-paying is.6$ an compls on τ6chang ay. hus Compaisons bwn opions pics bfo an af hi unlying sock bing iin-pai a awn in figu 3a an figu 3b. Jan.9 c.4 Cu of Paial isibuion Cu of sampl flogam Figu Paial isibuion fiing fo MCF a Compaison of call opion pics b Compaison of pu opion pics Cu of F pic Cu of B- pic Figu Compaison of opions pics on MF Compaisons bwn opions pics bfo an af h unlying sock bing capial-spli. uppos sik pic of opion is 5 aio of capial-spliing is k m m.96 an compls on τ8chang ay. hus Compaisons bwn opions pics bfo an af hi unlying sock bing capial-spli a awn in figu 4 a an figu 4b. 3 Compaisons bwn opions pics bfo an af h unlying sock bing iin-pai an capial-spli. uppos sik pic of opion is 65 iin-paying is.6$ aio of capial-spliing is k m m.96 an compls on τ9chang ay. hus Compaisons bwn opions pics bfo an af hi unlying sock bing iin-pai an capial-spli a awn in figu 5a an figu 5b. W s fom h ampls abo fo any cas h call opion shoul b cu bfo h unlying sock bing iin-pai o capial-spli if n an h pu opion shoul b cu af h unlying sock bing iin-pai o capial-spli if n. 7

8 Now w ha iscuss h mols an mhos of opion picing whn unlying sock is iin-pai o capial-spli fo on im. h iscussion is simila fo many ims. 6.6 τ τ 6 a Compaison of call opion pics b Compaison of pu opion pics Cu of F pic Cu of B- pic Figu 3 Compaisons of opions pics bfo an af MF bing iin-pai 5 k.96 τ 8 5 k.96 τ 8 a Compaison of call opion pics b Compaison of pu opion pics Cu of F pic Cu of B- pic Figu 4 Compaisons of opions pics bfo an af MF bing capial-spli 8 Conclusions Bas on h sucu mol fo opion picing 7] his pap gis h sucu mols fo opions picing on h cass which unlying sock is iin-pai capial-spli an iin-pai an capial-spli. hs mols a suiabl o Amican o Euopan opion. h call opion shoul b cu bfo h unlying sock bing iin-pai o capial-spli if 8

9 n an h pu opion shoul b cu af h unlying sock bing iin-pai o capial-spli if n. is a aluabl conclusion. I is inoscula o h pacic in financial aing an usful o as in financial mak. Fo oiginal F sucu mol o i on iin-paying an capial-spliing h is an assumpion in scion i.. h a no ansacion coss o as in aing. In fac h picing fomulas 4 6 an a also igh whn h ansacion coss o as in aing a subac fom hm. Finally h ampls abou opions picing on sock of MF a gin which inclu ha h sock of MF is iin-pai capial-spli an iin-pai an capial-spli. Compaing h pics on F sucu mols an hos on Black-chols mols h pics on F sucu mols a aional an b. o h sachs in his pap suppo h F sucu mol fo opions picing fuh k.96 τ k.96 τ 9 a Compaison of call opion pics b Compaison of pu opion pics Cu of F pic Cu of B- pic Figu 5 Compaisons of opions pics bfo an af MF bing iin-pai an capial-spli Rfncs ] Black F. an M.chols. h Picing of Opions an Copoa Liabiliis. Jounal of Poliical Economics ] Mon R.C. Opion Picing whn Unlying ock Runs a isconinuous. Jounal of Financial Economics ] hap W.F. InsmnM]. Pnic-Hall Inc ] Co J.C. an.a.ross Valuaion of Opions fo Alnai ochasic Pocss]. Jounal of Poliical Economics ] Whaly R. On h Valuaion of Amican Call Opions on ocks wih Known iins]. Jounal of Financial Economics ] Gsk R. an R. Roll. On Valuing of Amican Call Opions wih h Black-chols Euopan Fomula. Jounal of Financ ] JohnsonH.E. An Analyic Appoimaion o h Amican Pu Pic. Jounal of Financial an Quaniai Analysis Mach ] Gsk R. an JohnsonH.E. h Amican Pu Valu Analyically]. Jounal of Financ cmb ] Baon-AsiG. an R.E.Whaly. Efficin Analyic Appoimaion of Amican Opion Valus. Jounal of Financ 987 4Jun 3-3. ] MacMillanL.W. Analyic Appoimaion fo h Amican Pu Opion. Aancs in Fuus an Opion Rsach ] Ca P. R. Jaow an R. Mynni. Alnai Chaacizaions of Amican Pu Opions. Mahmaical Financ

10 ] aplon R.. an M. G. ubahmanyam. h Valuaion of Amican Opion wih ochasic Ins Ras: A Gnalizaion of h Gsk-Johnson chniqu Jounal of Financ ] John C. Hull. Opions Fuus an Oh iais 4h. Pnic Hall Inc ] Biys E. Opions Fuus an Eoic iais Jhon Wily & ons Inc ] Richn P. an R.o. Picing Opions Un Gnaliz GARCH an ochasic Volailiy Pocsss Jounal of Financ ] ai F. an G. Ji A Nw Kin of Picing Mol fo Commoiy an Esimaing Ins ysm fo Pic cuiy Chins Jounal of Managmn cinc ] ai F. an Z.F.Qin. F ucu Mols fo Opions Picing ICFAI Jounal of Appli Economics 5 fohcoming.